## Vacuum-induced jitter in spatial solitons

Optics Express, Vol. 3, Issue 5, pp. 171-179 (1998)

http://dx.doi.org/10.1364/OE.3.000171

Acrobat PDF (201 KB)

### Abstract

We perform a calculation to determine how quantum mechanical fluctuations influence the propagation of a spatial soliton through a nonlinear material. To do so, we derive equations of motion for the linearized operators describing the deviation of the soliton position and transverse momentum from those of a corresponding classical solution to the nonlinear wave equation, and from these equations we determine the quantum uncertainty in the soliton position and transverse momentum. We find that under realistic laboratory conditions the quantum uncertainty in position is several orders of magnitude smaller the classical width of the soliton. This result suggests that the reliability of photonic devices based on spatial solitons is not compromised by quantum fluctuations.

© Optical Society of America

## 1. Introduction

1. A pedagogical discussion of self-action effects including self-trapping and optical solitons is presented in Chapter 6 of R. W. Boyd, *Nonlinear Optics* (Academic, San Diego, 1992). See also E. M. Nagasako and R. W. Boyd, in *Amazing Light, A volume dedicated to Charles Hard Townes on his 80th Birthday*, edited by R. Y. Chiao (Springer, New York, 1996).

3. R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett.13, 479 (1964). [CrossRef]

5. E. L. Dawes and J. H. Marburger, Phys. Rev.179, 862 (1969); See also J. H. Marburger, Prog. Quantum Electron.4, 35 (1975). [CrossRef]

6. Y. Silberberg, Opt. Lett.15, 1282 (1990). [CrossRef] [PubMed]

7. J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. SmithJ. Opt. Soc. Am. B8, 1290 (1991). [CrossRef]

10. B. Luther-Davies and X. Yang, Opt. Lett.17, 1755 (1992). [CrossRef] [PubMed]

11. S. Blair, K. Wagner, and R. McLeod, Opt. Lett.19, 1943 (1994). [CrossRef] [PubMed]

12. E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A55, 1412 (1997). [CrossRef]

13. J. D. Gordon and H. A. Haus, Opt. Lett.11, 665 (1986). [CrossRef] [PubMed]

14. P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamomoto, Nature, 365, 307 (1993). [CrossRef]

15. H. A. Haus and M. N. Islam, IEEE J. Quantum Electron.QE-21, 1172 (1985). [CrossRef]

16. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B7, 386 (1990). [CrossRef]

16. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B7, 386 (1990). [CrossRef]

## 2. Theoretical Formulation

*A*is defined according to the convention

*n̄*

_{2}= (

*n*

_{0}

*c*/4

*π*)

*n*

_{2}where

*n*

_{2}is the usual nonlinear refractive index defined such that Δ

*n*=

*n*

_{2}

*I*. We then find that the nonlinear Schrodinger equation (for the positive frequency part of the field) takes the standard [16

16. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B7, 386 (1990). [CrossRef]

*X*,

*θ*

_{0}is the soliton phase,

*p*

_{0}is the transverse soliton momentum, and

*X*

_{0}is the position of the soliton center.

^{(+)}which we treat quantum mechanically:

*t*is the response time of the nonlinear response and

*L*

_{y}is the thickness of the slab waveguide that confines the radiation in the

*y*direction. The form of the coefficient

*S*is obtained by requiring that the total electric field obey the standard field commutation relation and then reducing the problem to one transverse dimension and one frequency component, under the assumptions that the spectrum is essentially uniform over the frequency interval 1/Δ

*t*and that the field amplitude is essentially constant over the thickness

*L*

_{y}of the waveguide. By substituting expression (5) into the nonlinear Schrodinger equation (1) and linearizing in the perturbation, the evolution equation for the perturbation

16. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B7, 386 (1990). [CrossRef]

^{(+)}can be represented as the sum of contributions resulting from fluctuations in the four fundamental soliton degrees of freedom:

*j*=

*N*,

*θ*,

*p*and

*X*and are given explicitly by

*X,Z*) as well as their adjoints

_{j}

^{(+)}(

*X,Z*) (see Appendix A for a description of the adjoint operation in the present context), we can now invert the expansion of the perturbation operator

^{(+)}to give the individual fluctuation operators as follows :

*N̂*

_{0}and Δ

*θ̂*

_{0}and the pair Δ

*X̂*

_{0}and

*N*

_{0}Δ

*p̂*

_{0}form conjugate operator pairs, and lead to uncertainty relations between the conjugate quantities. We shall see below (Eq. (22)) that the parameter

*N*

_{0}is typically smaller than unity, with the consequence that uncertainty relation expressed in terms of position Δ

*X̂*

_{0}and momentum Δ

*p̂*

_{0}(without the factor of

*N*

_{0}) predicts a quantum mechanical uncertainty increased over the single particle case by a factor of 1/

*N*

_{0}.

*Z*within the interaction region can be determined similarly. For simplicity, we consider only the case of a soliton propagating along the

*Z*axis, that is, we set

*p*

_{0}and

*X*

_{0}equal to zero. We then obtain the results

*X̂*

_{0}and

*N*

_{0}Δ

*p̂*

_{0}obey quantum mechanical commutation relations. Moreover, we show in Appendix B that the soliton position fluctuation operator Δ

*X̂*and momentum fluctuation operator Δ

*p̂*obey the equations

*Z*→

*t*. The factor of 2 in the second of Eqs. (14) results as a consequence of the particular conventions used in this calculation. These equations can be integrated to express the output fluctuations in terms of the input fluctuations as follows

*Z*. As before, we consider the case of a soliton propagating along the

*Z*axis, that is, we set

*p*

_{0}and

*X*

_{0}equal to zero. Then through use of Eqs. (11) we find that the initial uncertainties in the soliton position and momentum, that is, the uncertainties at the entrance plane (

*Z*= 0) are given by

*Z*in the material is given by

*w*along the

*z*axis is

*p*

_{0},

*X*

_{0}, and

*θ*

_{0}set equal to zero, we find that the parameter

*N*

_{0}of the normalized solution is given by

## 3. Discussion and Conclusions

*z*≪

*πkw*

^{2}/2. For any value of

*z*that satisfies this condition, including

*z*= 0, the uncertainty in position is dominated by the first term. Under these conditions, the fractional uncertainty in soliton position will be given approximately as

*k*

_{0}= 1.3 × 10

^{5}cm

^{-1}, corresponding to a wavelength of 0.5

*μ*m, and that

*w*and

*L*

_{y}are both of the order of 50

*μ*m. Material parameters enter into the calculation as the ratio

*n*

_{2}/Δ

*t*. This ratio tends to be nearly constant for most nonlinear optical materials, because materials that display a large nonlinear respone tend to be slow. One of the largest values of this ratio occurs for the conjugated polymers, for which

*n*

_{2}can be as large as 3×10

^{-12}cm

^{2}/W and for which the response time Δ

*t*is believed to be as short as 10 fs. We then find that (Δ

*x*)

_{rms}/

*w*≈ 10

^{-3}, which is at best barely measurable in the laboratory. Of course, the predicted fractional uncertainty in soliton position would be larger if calculated for a material with a larger value of

*n*

_{2}/Δ

*t*. Note also that Eq. (23), if extrapolated to values of z outside of its demonstrated limits of validity, predicts a linear increase in the fractional uncertainly in soliton position with increasing propagation distance z. It is not clear at present what the accurately predicted uncertainty would be under these conditions.

^{3}and consequently is barely experimentally observable. However, the fractional uncertainty could be considerably larger through use of materials with a larger nonlinear respponse or under conditions that lie outside of the validity of the present theory. These larger fractional encertainties in soliton position could have considerable importance for the construction of optical switching devices that rely on the properties of spatial solitons.

## 4. Acknowledgments

## Appendix A

*X, Z*) and

*X, Z*) to Eq. (7) must obey the equations

*L*is defined as

_{i}

^{(+)}may be considered to be the adjoint of

*X, Z*) and

*X, Z*) would have the form

*f*

_{i}(

*X*) is the eigenfunction and

*E*

_{i}is the eigenvalue. If this form is substituted into Eq. (29) we find that the equation is satisfied for

*E*

_{1}≠

*E*

_{2}if

## Appendix B

*X, Z*), we find that Δ

*p̂*(

*Z*) can be written as

*X̂*(

*Z*) as defined by last of Eqs. (13). Through use of Eqs. (10) and (28) we find that

*X̂*(

*Z*) can be written as

## References and links

1. | A pedagogical discussion of self-action effects including self-trapping and optical solitons is presented in Chapter 6 of R. W. Boyd, |

2. | G. A. Askar’yan, Sov. Phys. JETP15, 1088 (1962). |

3. | R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett.13, 479 (1964). [CrossRef] |

4. | V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP34, 62 (1972). |

5. | E. L. Dawes and J. H. Marburger, Phys. Rev.179, 862 (1969); See also J. H. Marburger, Prog. Quantum Electron.4, 35 (1975). [CrossRef] |

6. | Y. Silberberg, Opt. Lett.15, 1282 (1990). [CrossRef] [PubMed] |

7. | J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. SmithJ. Opt. Soc. Am. B8, 1290 (1991). [CrossRef] |

8. | M. Shalaby and A. Barthelemy, Opt. Comm.94, 341 (1992). [CrossRef] |

9. | A. Villeneuve, J. S. Aitchison, J. U. Kang, P. G. Wigley, and G. I. Stegeman, Opt. Lett.19, 761 (1994). [CrossRef] [PubMed] |

10. | B. Luther-Davies and X. Yang, Opt. Lett.17, 1755 (1992). [CrossRef] [PubMed] |

11. | S. Blair, K. Wagner, and R. McLeod, Opt. Lett.19, 1943 (1994). [CrossRef] [PubMed] |

12. | E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A55, 1412 (1997). [CrossRef] |

13. | J. D. Gordon and H. A. Haus, Opt. Lett.11, 665 (1986). [CrossRef] [PubMed] |

14. | P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamomoto, Nature, 365, 307 (1993). [CrossRef] |

15. | H. A. Haus and M. N. Islam, IEEE J. Quantum Electron.QE-21, 1172 (1985). [CrossRef] |

16. | H. A. Haus and Y. Lai, J. Opt. Soc. Am. B7, 386 (1990). [CrossRef] |

17. | Although |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(270.5530) Quantum optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 5, 1998

Revised Manuscript: May 5, 1998

Published: August 31, 1997

**Citation**

Elna Nagasako, Robert Boyd, and Girish Saran Agarwal, "Vacuum-induced jitter in spatial solitons," Opt. Express **3**, 171-179 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-5-171

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### References

- A pedagogical discussion of self-action effects including self-trapping and optical solitons is presented in Chapter 6 of R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992). See also E. M. Nagasako, R. W. Boyd, in Amazing Light, A volume dedicated to Charles Hard Townes on his 80th Birthday, edited by R. Y. Chiao (Springer, New York, 1996).
- G. A. Askar'yan, Sov. Phys. JETP 15, 1088 (1962).
- R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964). [CrossRef]
- V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
- E. L. Dawes and J. H. Marburger, Phys. Rev. 179, 862 (1969); See also J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975). [CrossRef]
- Y. Silberberg, Opt. Lett. 15, 1282 (1990). [CrossRef] [PubMed]
- J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. Smith, J. Opt. Soc. Am. B 8, 1290 (1991). [CrossRef]
- M. Shalaby and A. Barthelemy, Opt. Comm. 94, 341 (1992). [CrossRef]
- A. Villeneuve, J. S. Aitchison, J. U. Kang, P. G. Wigley, and G. I. Stegeman, Opt. Lett. 19, 761 (1994). [CrossRef] [PubMed]
- B. Luther-Davies and X. Yang, Opt. Lett. 17, 1755 (1992). [CrossRef] [PubMed]
- S. Blair, K. Wagner, and R. McLeod, Opt. Lett. 19, 1943 (1994). [CrossRef] [PubMed]
- E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A 55, 1412 (1997). [CrossRef]
- J. D. Gordon and H. A. Haus, Opt. Lett. 11, 665 (1986). [CrossRef] [PubMed]
- P. D. Drummond R. M. Shelby, S. R. Friberg, and Y. Yamomoto, Nature 365, 307 (1993). [CrossRef]
- H. A. Haus and M. N. Islam, IEEE J. Quantum Electron. QE-21, 1172 (1985). [CrossRef]
- H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990). [CrossRef]
- Although c has the fixed value -1/2 using the present conventions, we retain c in our formulas for more ready comparisons of our results with those obtained using different normalization conventions.

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